Gnedenko-Raikov’s Theorem Fails for Exchangeable Sequences

George Stoica, Deli Li

Department of Mathematical Sciences, Lakehead University, Thunder Bay, Canada.
Department of Mathematical Sciences, University of New Brunswick, Saint John, Canada.

DOI: 10.4236/am.2012.312A277                

Abstract

We study the connection between the central limit theorem and law of large numbers for exchangeable sequences, and provide a counterexample to the Gnedenko-Raikov theorem for such sequences.

Keywords

Exchangeable Sequences; Central Limit Theorem; Weak Law of Large Numbers

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Gut, “Gnedenko-Raikov’s Theorem, Central Limit Theory, and the Weak Law of Large Numbers,” Statistics and Probability Letters, Vol. 76, No. 17, 2006, pp. 1935-1939. doi:10.1016/j.spl.2006.04.042
[2]	M. Klass and H. Teicher, “The Central Limit Theorem for Exchangeable Random Variables without Moments,” Annals of Probability, Vol. 15, No. 1, 1987, pp. 138-153. doi:10.1214/aop/1176992260
[3]	G. Stoica and D. Li, “On the Kolmogorov-Feller Law for Exchangeable Random Variables,” Statistics and Probability Letters, Vol. 80, No. 9-10, 2012, pp. 899-902. doi:10.1016/j.spl.2010.01.025
[4]	G. Stoica, “An Extension of the Weak Law of Large Numbers for Exchangeable Sequences,” Acta Applicandae Mathematicae, Vol. 109, No. 3, 2010, pp. 759-763. doi:10.1007/s10440-008-9344-x
[5]	M. V. Kruglov, “A Generalization of the Weak Law of Large Numbers,” Stochastic Analysis and Applications, Vol. 29, No. 4, 2011, pp. 674-683. doi:10.1080/07362994.2011.581099
[6]	X. Jiang and M. G. Hahn, “Empirical Central Limit Theorems for Exchangeable Random Variables,” Statistics and Probability Letters, Vol. 59, No. 1, 2002, pp. 75-81. doi:10.1016/S0167-7152(02)00204-3
[7]	G. Stoica, “A Self-Normalized Weak Law of Large Numbers for Exchangeable Sequences,” Advances and Applications in Statistics, Vol. 25, No. 2, 2011, pp. 103-108.